x-o_ csx srn5 12T ! x l,cl Calculus I Final Exam Summer 2014 bc1 Name

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9x Calculus I 090 Final Exam Summer 04 Name Instructions.

Answers unaccompa nied by work may Hot receive credit.

. (lfipts) Find the fr)llowing derivatives. Show your work below and circle your final answer.

1 9x Calculus I 090 Final Exam Summer 04 Name Instructions. Show all work and include appropriate explanations when necessary. Answers unaccompa nied by work may Hot receive credit. Please try to do all work iii the space provided and circle your final answers.. (lfipts) Find the fr)llowing derivatives. Show your work below and circle your final answer. (a) (4pts) D(x + 6) (b) (4pts)D.(sinxcosx). (Si )(_sit)l) csx srn5 (c) (4pts) D() lyx +qx (d) (4pts) D(sin (x 4 x)). (8pts) Compute the following limits. Answers may be values, ±oo, or DNE. Show your work below and circle your final answer. (a) (4pts) urn T s )(...4 )y. o a l.wcu.(c yosihii e. S (b) (4pts) lim )IL4A _ LIL4A. E+O x Xô bc! x xo_ l,cl

i). (6pts) Find the equation of the tangent line to the graph of

a) () iq(x) = f) =.i. at the point (, ). 4.

derivative of the function fqr) = 6: that is.

(lopts) Billy Joe wants to build a rectangular pig pen of area 8

of the enclosure are fence since the side of his cabin will form

What dimensions (labeled x and y in the picture below) should

2 i). (6pts) Find the equation of the tangent line to the graph of y = (x f) ()t 4 a) () iq(x) = f) =.i. at the point (, ). 4. (6pts) Use the definition of the derivative to compute the derivative of the function fqr) = 6: that is. compute f(x+h)f(x) inn h i v.. L jg I L kgo + 5. (lopts) Billy Joe wants to build a rectangular pig pen of area 8 square meters up against the side of his cabin; only three sides of the enclosure are fence since the side of his cabin will form a fourth wall (see picture below). What dimensions (labeled x and y in the picture below) should Billy Joe make the pen to use the least amount of fence? Minimize the perimeter P = y + x subject to the constraint xy = 8. Note: You must use calculus to get credit!! CABIN WALL M i XI )., Ag y F c. 0 Ct = ) x so

6. (Opts) Consider the function (a) (pts) Find (x).

f(x) = x Ox + 8 (c) (pts) Find the critical point(s) of f.

below: f(x) is at x =. CONCAVE UP VE DOWN.f ().

a local minimum, a local maximum, or neither. X ic 0 i. ve.lfla.

3 6. (Opts) Consider the function (a) (pts) Find (x). N O (b) (pts) Find f (x). f(x) = x Ox + 8 (c) (pts) Find the critical point(s) of f. Lcx&o Z (d) (pts) Fill in the blank by circling the cect aeelow: f(x) is at x. INCREASING ASING f?) (e) (pts) Fill in the blank by circling the correct answer below: f(x) is at x =. CONCAVE UP VE DOWN.f (). o (f) (4pts) Classify each critical point you found in part (c) as a local minimum, a local maximum, or neither. X ic 0 i. ve.lfla.cf ( & bt4(i (g) (pts) Find the inflection point(s) of f. (,o _ lôc.) f f Lz.. = I;, J (6pts) An object is thrown upwards off a 00 foot tall building. Its velocity after t seconds is given by v(t) t + feet per second. How far off the ground is the object after seconds? i% f c 4A Sv)L +) l+jc loo C SL),o ) () i() l()too

4 x 8. (6pts) Use the graph of y f(x) below to computej f(x) dx = I to7 _Lt yf(x) 4 9. (8pts) Find the following antiderivatives. Remember: +C! ) (a) (4pts) /(6x 5x +) dx c5l (b) (4pts) /cosxsinx dx Note: cos x is the same as (cos x). 0. (8ps) Evaluate the following definite integrals using the Fundamental Theorem of Calculus. (a) (4pts) /(x) dx (b) (4pts) I L r / x(x i) dx. u I (if (o). (6pts) Evaluate the Riemann sum for f(x) = x on the interval [,] using the partition of 4 subintervals of equal length with the sample points being the leftendpoints of each subinterval. Co, 0 i x Zj fig) xy 4

). (Opts) Consider the region R in the first quadrant bounded

Figure A below is a rough sketch of the region R.

the region R around the iaxis using the washer nouiod V ()4 rfr

. (b) (lopts) Find the volume of the solid obtained by rotating

the iaxis, and x = sketched in Figure B below.

5 ). (Opts) Consider the region R in the first quadrant bounded by y = Vx +, the iaxis, and x =. Figure A below is a rough sketch of the region R. (a) (lopts) Find the volume of the solid obtained by rotating the region R around the iaxis using the washer nouiod V ()4 rfr = ]r(f+z) K).. (b) (lopts) Find the volume of the solid obtained by rotating the region R around the yaxis using the method of cylindrical shells. V f 0 0 (ix) ç (x ) I I,. (8pts) Consider the region S bounded by the curves y = x. the iaxis, and x = sketched in Figure B below. Each integral below is the volume of a solid obtained by rotating S around a particular axis. Match the correct axis with the expression for volume by writing the appropriate letter in the blank provided. Each answer is used exactly once. D f x(x j f ) dx A. xaxis x((x + ) dx B. yaxis (x + )(x dx C. x = J ) 0ir(x) dxd.y= I Figure A Figure B 0

x 6 4. (6pts) Find the arc length of the parametric curve x = t + 4t + 9, y = + t and t 4.

(6pts) A metal rod is located between i = 0 and x on the xaxis (units are centimeters).

the location of the center of mass of the rod. 0 0 (. +e 0 (XZ/ = i Mr jxe) )() fr4s (%) 6.

the tank has the shape of a right triangle (see the picture below).

determine much work is required to pump the water out over the top edge of the tank.

6 x 6 4. (6pts) Find the arc length of the parametric curve x = t + 4t + 9, y = + t and t 4. )( Ui.:: (h j Xtft) I between t = 0 LJj4 LLt4*,) tf (t# ) 0 4 i (t )H (tti) 5. (6pts) A metal rod is located between i = 0 and x on the xaxis (units are centimeters). If the density of the rod at location x is given by p(x) = 4 grams per centimeter, find the location of the center of mass of the rod. 0 0 (. +e 0 (XZ/ = i Mr jxe) )() fr4s (%) 6. (l0pts) A tank is 4 feet tall, 4 feet wide, and 4 feet long; when viewed from the side, the tank has the shape of a right triangle (see the picture below). This tank is filled with water which has a density of 60 lbs/ft Use an integral to determine much work is required to pump the water out over the top edge of the tank. Give your answer in footpounds.. fti IL 4ff VoI tq I E tee F qok& 4) L Z b k ( k) ctl 4 j4 )o 4 W (ol(l)l Jo lo Jo Lf = (% (,Lj 6 (o)(z zc o +

Math Exam 03 Review

Math Exam 03 Review Math 10350 Exam 03 Review 1. The statement: f(x) is increasing on a < x < b. is the same as: 1a. f (x) is on a < x < b. 2. The statement: f (x) is negative on a < x < b. is the same as: 2a. f(x) is on